I've started my graduate school applications for the next year. When I try to provide my university information I fill in a form with my grades. Since I am doing an Msc course this year I have not yet received any grades. I can't follow through the application without them though. Should I just insert 0 and explain that later?
The best way to do this would be to insert "n/a" into the field for your grades, or simply write "not available yet" as CaptainEmacs suggested. In the event that you cannot write a non-numerical answer (as was the case when I applied to grad school), a "0" should suffice that it is not your actual grade. Your application will likely require preliminary transcripts (and eventually official transcripts), so I would not worry about not being able to report your official grade, especially since this is a common problem.
I would avoid writing this issue into your statement of purpose, since your SoP should be relatively short and to the point, focusing on your research and other accomplishments. While I would definitely include your MSc course in your statement, I would not mention anything relating to the application itself. Often universities have a graduate coordinator for any given department, so I suggest reaching out to them if you have questions about your application.
You should definitely explain your situation in your personal letter/CV that you are currently taking a master's course, and provide some details on what classes you're taking. This helps giving some insight into what you're doing even though you don't have any grades yet. Sometimes universities can ask for preliminary grades, so it is best to contact the graduate admissions officer and ask for details, especially since it can vary between universities.
I think the safest and most reasonable thing that you could do is to make a phone call to the department that you are applying to as soon as possible and inform them of your situation. They will tend to have an answer because of the volume of applications they have taken in over the years, odds are, they've come across "you" more than a handful of times.